Such phenomena should be observable in all ELBM schemes with the
classical entropies: there exists a vicinity of the equilibrium
where the entropic involution is possible but for some areas of
non-equilibrium distributions there exists no non-trivial root of
equation (REF ). A collision which preserves
entropy does not exist for this area. Therefore, for the regimes
close to equilibrium (the vicinities \(A\)  of equilibria, Figs 1,2),
ELBM schemes guaranty the precise balance of the entropy and for
more nonequilibrium regimes, when at some sites the distribution
belonges to sets \(B\) , ELBM schemes work as limiters
[1]}. with additional dissipation. It is necessary for
any complete definition of an ELBM algorithm to prescribe what to do
when the involution is not possible. A reasonable choice would be to
over-relax the maximum amount possible while maintaining positive
population values. Such a technique is independently in use as a
stabilizer for lattice Boltzmann schemes, sometimes called the
`positivity limiter' [2]}, [1]}, [4]}, [5]}, [6]}. An effect
of this operation is a local increase in viscosity/entropy
production. Hence, if an ELBM were to apply such a scheme it would
necessarily break the proper entropy balance. In this sense, ELBM
belongs to a large family of add-ons that regularise LBM by the
management of the addtional dissipation [7]}.
